I love how many different ways there are of seeing this pattern. I will try to explain what's going on in this picture:

In brown: a student noticed that the number of snowflakes in the top row of each pattern was the same as the step number and that each row below it had one more. So step 43 would be 43 + 44 + 45. Generally this would be n + (n + 1) + (n + 2) = 3n + 3 (sadly, I do not have that written anywhere).

In pink: a student noticed that each step has 3 snowflakes added on the diagonal. Working backwards to step 0, there would be 3 snowflakes so that rule here is 3 + 3n.

In black: some looked at the number of snowflakes, not at the pattern and again worked backward to get the starting value.

In blue: same as pink, but the pattern is growing by adding one column of 3 each time.

In orange: I showed them this one - we can move one of the single snowflakes to form a rectangle. For step 1 the dimensions are 3 by 2, for step 2 the dimensions are 3 by 3, for step 3 the dimensions are 3 by 4. Generalizing to step n, we would have a rectangle with dimensions 3 by (n + 1), and 3(n + 1) = 3n + 3.

So much good stuff in here!

Next it was time for the quiz. I tried to make it as stress-free as possible. I told them they could use their exercise books and worksheets, along with the graphing calculators (and instructions sheet). I also told them they could have as much time as they needed. We got the desks into rows and they got going. There were the usual questions along the way about how to use the graphing calculator, but others too when they were stuck. I think that developing skills around using the tools available to them is really important and not something that we do very explicitly. Knowing that they are stuck on a quadratic question should lead them to open their book to look at a quadratic example - what should the graph look like? what patterns should be expected in the data? how can I find other points?, but many just sat and stared for a while. I encouraged them as I know they know far more than they had so far shown me. They need to learn to ask themselves what they can do to start or continue a question, instead of leaving it blank (not allowed in my class) or immediately asking me. That resourcefulness is something many have not practiced much so I will attempt to help them help themselves more as we progress through the semester. Confidence in math comes not only from knowing how to answer a question, but knowing strategies to use when they don't know how to answer a question.

Many did not finish the quiz today so I will have them continue tomorrow. I want to see their best work and I will not let time be a barrier to that.

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